# Exceptional biases in counting primes over functions fields

## TL;DR

This paper investigates the rarity of certain biases in prime distributions over function fields, showing they become negligible as the size of the finite field grows, using advanced sieve and geometric methods.

## Contribution

It introduces new bounds demonstrating the vanishing probability of three types of prime biases in large finite fields, improving previous results by Kowalski.

## Key findings

- Biases occur with probability tending to zero as q increases
- New bounds improve upon Kowalski's earlier results
- Uses advanced sieve methods and arithmetic geometry techniques

## Abstract

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.